Extending monads along dense functors Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$.
$\require{AMScd}$
\begin{CD}
    \mathsf A @>T>> \mathsf A\\
    @V j V V @VV j V\\
    \mathsf B @>>\text{lan}_j(jT)> \mathsf B
\end{CD}

Is it possible to equip $\text{lan}_j(jT)$ with a monad structure?

Prop. 1 When $\text{lan}_j(jT)$ preserve itself, the answer is yes. Moreover, if the monad $T$ is idempotent, so is $\text{lan}_j(jT)$.
Sketch of Proof. The map $\text{lan}_j(j \circ \_): \text{End}(\mathsf A) \to \text{End}(\mathsf B)$  is strong monoidal on the full subcategory spanned by the monad $T$ and $1_{\mathsf A}$. Indeed $\text{lan}_j(jT^2) \cong \text{lan}_j(jT)^2,$ because $$\text{lan}_j(jT^2) \cong  \text{lan}_j(\text{lan}_j(jT)jT) \cong \text{lan}_j(jT) \circ \text{lan}_j(jT) = \text{lan}_j(jT)^2. $$
(Observe that part of the functoriality is due to the fact that $j$ is dense). In particular, it maps monoid object to monoid objects.
Cor. 2 When $j$ is the Yoneda embedding, $\text{lan}_j(jT)$ admits a monad structure.
Proof. By the universal property of the presheaf construction, $\text{lan}_j(jT)$ is cocontinous, thus preserve every (pointwise) Kan extension, in particular, itself. 
Cor. 3 Given a locally $\lambda$-presentable category $\mathsf{A}$ and a  $\lambda$-colimit preserving monad $T: \mathsf{A}_{\lambda} \to \mathsf{A}_{\lambda}$ (the full subcategory of $\lambda$-presentable objects), then there exists an extension $\mathsf{T}: \mathsf{A} \to \mathsf{A}$ admitting a monad structure. Moreover, $\mathsf{T}$ is cocontinuous.
Proof. Similar to Cor. 2.
Cor. 4 When $\text{lan}_j(jT)$ is absolute, $\text{lan}_j(jT)$ admits a monad structure.

  
*
  
*Is the argument in Prop. 1 correct?
  
*How natural is the hypothesis that $\text{lan}_j(jT)$ preserve itself?
  
*Are there natural or known assumption under which it is possible to equip $\text{lan}_j(jT)$ with a monad structure?
  

 A: This is only a partial answer to your questions, but one that I imagine recovers all the cases of interest (in particular it gives a conceptual reason why your examples hold when $j$ is the Yoneda embedding and when $j$ is the inclusion $A_\lambda \hookrightarrow \mathbf{Ind}(A_\lambda)$). Suppose that $j$ is well-behaved in the sense of Altenkirch–Chaptman–Uustalu: i.e. $j : A \to B$ admits left Kan extensions along functors $A \to B$, and is fully faithful, dense, and the nerve $N_j$ preserves left Kan extensions along $j$. (Such left Kan extensions will always be admitted when $A$ is small and $B$ is cocomplete.) Then $j$ is the inclusion of $A$ under some class $\Phi$ of weighted colimits (Theorem 8.11 of Szlachányi's On the tensor product of modules over skew monoidal actegories), and conversely.
In this case, there is a lax idempotent 2-monad $(\mathbf T, \mu, \eta)$ on $\mathbf{CAT}$ for which $\eta_A$ coincides with $j$ (up to equivalence): this follows from Lack–Kelly's On the monadicity of categories with chosen colimits. By general properties of lax idempotent 2-monads, the action of the 2-functor $\mathbf T$ on functors $f \colon A \to B$ is given by $\mathrm{Lan}_{\eta_A} (\eta_B \circ f)$. Consequently, since 2-monads preserve monads, if $T : A \to A$ has the structure of a monad on $A$, then $\mathbf T(T) \colon \mathbf T(A) \to \mathbf T(A)$ has the structure of a $\Phi$-cocontinuous monad on $\mathbf T(A)$.
You recover your examples for $j$ the Yoneda embedding taking the class of all small weights, and for $j$ the inclusion $A_\lambda \hookrightarrow \mathbf{Ind}(A_\lambda)$ taking the class of filtered weights. All of this works just as well in the enriched setting.
